Part IV: Electromagnetism & Fields
Special relativity was born from electromagnetism. Maxwell's equations are already relativistic—they're the same in all inertial frames. This part reveals the deep unity of electric and magnetic fields and introduces the tensor formulation of electromagnetism.
Part Overview
Electric and magnetic fields are not separate entities—they're components of a single electromagnetic field tensor . What one observer sees as a pure electric field, another moving observer sees as a combination of electric and magnetic fields. This part develops the covariant formulation of electromagnetism, showing how Maxwell's equations take a beautiful tensor form and introducing the Lagrangian formulation that leads to quantum field theory.
Key Topics
- • The electromagnetic field tensor and its components
- • Covariant form of Maxwell's equations:
- • How electric and magnetic fields transform between frames
- • Four-current and charge conservation
- • Electromagnetic invariants: and
- • Lagrangian formulation and the action principle
- • Path to general relativity: the equivalence principle
6 chapters | Unity of E and B fields | Maxwell meets Einstein
Chapters
Chapter 1: Electromagnetic Field Tensor
The field tensor unifies electric and magnetic fields. Its components are and . The antisymmetric tensor structure: . The dual tensor and the electromagnetic invariants. How the tensor transforms under Lorentz boosts.
Chapter 2: Covariance of Maxwell's Equations
Maxwell's four equations reduce to two tensor equations: (inhomogeneous) and (homogeneous). The Bianchi identity. Gauge invariance and the four-potential : .
Chapter 3: E and B Field Transformations
How electric and magnetic fields transform between frames. Parallel and perpendicular components. A pure electric field in one frame appears as a combination of E and B in another. The magnetism of a current-carrying wire is just electrostatics plus relativity. Worked examples.
Chapter 4: Four-Current and Charge Conservation
The four-current unifies charge density and current density. Continuity equation as a four-divergence: . Charge conservation is a consequence of gauge invariance. How current density transforms. The Lorentz force in covariant form: .
Chapter 5: Lagrangian Formulation
The electromagnetic Lagrangian density: . Deriving Maxwell's equations from the action principle. Euler-Lagrange equations for fields. The energy-momentum tensor and conservation of energy-momentum. Noether's theorem and symmetries.
Chapter 6: Path to General Relativity
Special relativity handles inertial frames; general relativity handles accelerated frames and gravity. The equivalence principle: acceleration is indistinguishable from gravity. Why gravity must curve spacetime. The metric tensor generalizes the Minkowski metric. Geodesics in curved spacetime. Preview of Einstein's field equations.
Course Navigation
Prerequisites:
- • Part I: Spacetime Foundations
- • Part II: Lorentz Transformations
- • Part III: Relativistic Mechanics
- • Basic electromagnetism (Maxwell's equations)
- • Tensor notation and index manipulation